Question

The value of the expression ${}^{47}{C}_4+\sum _{j=1}^5{{}^{52-j}C}_3$ is

• A. ${}^{51}{C}_4$
• B. ${}^{52}C_4$
• C. ${}^{52}C_3$
• D. ${}^{53}C_4$

Answer 1

The correct option is B

${}^{52}C_4$

Explanation for correct option:

${}^{47}{C}_4+\sum _{j=1}^5{{}^{52-j}C}_3={}^{47}{C}_4{+}^{51}{C}_3{+}^{50}{C}_3{+}^{49}{C}_3{+}^{48}{C}_3{+}^{47}{C}_3$

We know that${}^{\mathit{n}}{\mathit{C}}_{\mathit{r}}{\mathbf{+}}^{\mathit{n}}{\mathit{C}}_{\mathit{n}\mathbf{-}\mathit{r}}{\mathbf{=}}^{\mathit{n}\mathbf{+}\mathbf{1}}{\mathit{C}}_{\mathit{r}}$

$$\begin{gathered} \therefore {}^{47}{C}_4{+}^{51}{C}_3{+}^{50}{C}_3{+}^{49}{C}_3{+}^{48}{C}_3{+}^{47}{C}_3 \\ =\left({}^{47}{C}_4{+}^{47}{C}_3\right){+}^{48}{C}_3{+}^{49}{C}_3{+}^{50}{C}_3{+}^{51}{C}_3 \\ =\left({}^{48}{C}_4{+}^{48}{C}_3\right){+}^{49}{C}_3{+}^{50}{C}_3{+}^{51}{C}_3\left[{\because }^{\mathit{n}}{\mathit{C}}_{\mathit{r}}{\mathbf{+}}^{\mathit{n}}{\mathit{C}}_{\mathit{n}\mathbf{-}\mathit{r}}{\mathbf{=}}^{\mathit{n}\mathbf{+}\mathbf{1}}{\mathit{C}}_{\mathit{r}}\right] \\ =\left({}^{49}{C}_4{+}^{49}{C}_3\right){+}^{50}{C}_3{+}^{51}{C}_3\left[{\because }^{\mathit{n}}{\mathit{C}}_{\mathit{r}}{\mathbf{+}}^{\mathit{n}}{\mathit{C}}_{\mathit{n}\mathbf{-}\mathit{r}}{\mathbf{=}}^{\mathit{n}\mathbf{+}\mathbf{1}}{\mathit{C}}_{\mathit{r}}\right] \\ =\left({}^{50}{C}_4{+}^{50}{C}_3\right){+}^{51}{C}_3\left[{\because }^{\mathit{n}}{\mathit{C}}_{\mathit{r}}{\mathbf{+}}^{\mathit{n}}{\mathit{C}}_{\mathit{n}\mathbf{-}\mathit{r}}{\mathbf{=}}^{\mathit{n}\mathbf{+}\mathbf{1}}{\mathit{C}}_{\mathit{r}}\right] \\ {=}^{51}{C}_4{+}^{51}{C}_3\left[{\because }^{\mathit{n}}{\mathit{C}}_{\mathit{r}}{\mathbf{+}}^{\mathit{n}}{\mathit{C}}_{\mathit{n}\mathbf{-}\mathit{r}}{\mathbf{=}}^{\mathit{n}\mathbf{+}\mathbf{1}}{\mathit{C}}_{\mathit{r}}\right] \\ {=}^{52}{C}_4\left[{\because }^{\mathit{n}}{\mathit{C}}_{\mathit{r}}{\mathbf{+}}^{\mathit{n}}{\mathit{C}}_{\mathit{n}\mathbf{-}\mathit{r}}{\mathbf{=}}^{\mathit{n}\mathbf{+}\mathbf{1}}{\mathit{C}}_{\mathit{r}}\right] \end{gathered}$$

Hence, option (B) is correct.